The electronic energy bands of the alkali metals and metallic beryllium By J. F. Cornwell Department of Mathematics, Imperial College, London ( Communicated by H. Jones, F.R.S.—Received 10 December 1960) The applicability of various electronic energy band interpolation schemes to the alkali metals and metallic beryllium is discussed. An ‘ Z-dependent ’ pseudo-potential method using only a very small number of plane waves in the expansion of the valence electron wave functions is then applied to these metals. The density of states curves and the Fermi surfaces are calculated. It is found that for lithium the Fermi surface may touch the Brillouin zone boundaries. The calculated and observed low temperature electronic specific heat coef ficients and valence band widths are compared.
552 J. F. Cornwell 2. Previous interpolation schemes Three energy band interpolation schemes have previously been proposed. The first was by Slater & Koster (1954), whose scheme was based on the tight binding approximation. This method is not really appropriate for the valence bands of non transition metals. Allen (1955) proposed a scheme based on the orthogonalized plane wave method, suggesting that the Fourier coefficients of the crystal potential and the matrix elements of the Hamiltonian between core functions be treated as disposable parameters. A considerably simpler scheme was proposed by Phillips 1 <1958, 1959) in an application to diamond, silicon, and germanium. Phillips took account of the orthogonality with core electron functions by introducing a repulsive potential. The first two (three in the case of germanium) independent Fourier coefficients of the pseudo-potential obtained by combining this repulsive potential j with the crystal potential were taken as disposable parameters, the values of which 1 were chosen by fitting to the energy levels at the symmetry points of the B.Z. that had previously been calculated by other authors. The energy levels were then j obtained for points along the symmetry axes using about 27 plane waves in the expansion of the valence electron wave functions. The results were remarkably good, | the energy levels at the symmetry points being reproduced quite well. In a later paper, Phillips & Kleinmann (1959) have dealt in more detail with the theory | j behind this form of the pseudo-potential method. Antoncik (1954, 1959a) and Callaway (19586) have applied similar forms of pseudo-potentials to the calculation of the energy levels at the symmetry points of the B.Z. for the alkali metals. Although differing in detail, these three calculations j & j were all based on the potential l^(r) proposed by Hellmann Kassatotsehkin (1936), F„(r) being given by Fp(r) = - 2/r + A exp (- where A and /? are parameters.
The alkali metals and metallic beryllium 553 have to be made orthogonal to the core electron wave functions. In this approach therefore, the repulsive pseudo-potential is only introduced in the calculation of the 5-like levels, the p-like and d-like levels, etc. being calculated directly from the crystal potential. According to the calculations of Glasser & Callaway (1958), using the notation of Bouckaert, Smoluchowski & Wigner (1936), the lowest state at the centre of the B.Z., 71, is a state, an 5-like state, while the lowest state at the point N is an state, a p-like state. Along the axis the 5-like part of the valence electron wave decreases continuously, vanishing at N. The pseudo-potential methods outlined so far cannot take account of this continuous variation, and for this reason a more sophisticated method taking explicit account of the symmetries has to be used. For heavier metals which have core electrons in states of several different symmetries an *Z-independent ’ pseudo-potential, such as that proposed by Phillips, is probably not such a poor approximation as it would be for lithium and beryllium. Harrison (1960 a) has applied such a scheme to aluminium, and has also (Harrison 19606) applied a very considerably simplified form of it to other poly valent metals.
554 J. F. Cornwell Here Q is the volume of the unit cell and Nj is a normalizing constant. Antoncik has then shown that f tfHvXi&T = NINE'S a*taql{y^Sm + Ffk* - ly + (F,ep.)M}l, (*) Jo , p, a where (frep.)pa = ^ S exP { — i(kp — ka) • S (47r(2^ +1) ij(cos 0pq) mm j/= 1 l V x J{ - r2 d»*}. (5) In these equations V(kp - kg) is a Fourier coefficient of the crystal potential, is a Legendre polynomial, jt a spherical Bessel function, and dpq is the angle between the vectors kp and kQ. The summation over l is over all the states with angular momentum quantum number that occur in the ion core, and the summation over v is over all the atomic sites in the unit cell.
The alkali metals and metallic beryllium 555 (a) Lithium The energy levels at the symmetry points of the B.Z. of lithium have been calcu lated by a number of authors, probably the most accurate and extensive calculation being that of Glasser & Callaway (1958), who used the orthogonalized plane wave method and the corrected Seitz crystal potential (Kohn & Rostoker 1954). Owing to the non-vanishing of the normal derivative of the Seitz potential at the boundaries of the unit cell there is an estimated error in all but the lowest Tx level of ± 0-05 Ry (Callaway 1958a). As it appears that the touching or non-touching of the zone boundaries by the Fermi surface depends critically on the energies of the levels at the symmetry points assumed, in particular on the lowest level at N, two calcula tions were performed for lithium. In the first one, which will be called calculation , the lowest energy level at N was assumed to be that given by Glasser & Callaway, namely — 0*404 Ry. In the calculation B, this level was taken 0*05 Ry lower.
556 J. F. Cornwell a being a disposable parameter which was chosen by fitting to the lowest /\ level. For calculation A, a was taken as 3*17, for B it was taken as 2*84. It may be noted that for a free lithium atom, equation (6) is a good approximation when a = 2*69 (Morse, Young & Haurwitz 1935). Table 1 gives a comparison of the lower energy levels calculated by this method with those given by Glasser & Callaway. The most important are those that correspond to the first band. These, for lithium, are the states r x, N'ly P4 and H15.
The alkali metals and metallic beryllium 557 (6) Sodium The most extensive calculation of the energy levels at the symmetry points of the B.Z. for sodium is that of Howarth & Jones (1952), who used the cellular method. Hydrogenic analytical wave functions were chosen for the Is, 2s and 2 core electrons, three disposable parameters being involved, two for the s-like states and Table 2. Energy levels at the symmetry points of the B.Z. for sodium.
558 J. F. Cornwell free electrons with the same effective mass at the B.Z. centre. For all the alkali metals there is an ordinary saddle point of the type S2 (see VanHove 1953) at which makes N'(E) od as E -> EN from the left. A summary of the results of the calculation is given in table 4. Sodium also undergoes a martensitic transformation at low temperatures (Barrett 1948), and Martin (1960 a) has shown that the observed low temperature specific heat is dependent on the thermal history of the sample.
The alkali metals and metallic beryllium 559 (d) Rubidium and caesium There have been no accurate calculations of the energy levels at all the symmetry points of the B.Z. for rubidium. For caesium, the best calculation is probably that of Callaway & Haase (1957). As the band width is very narrow, the calculated dif ference between the lowest energy levels at and N being only 0*085 Ry (cf. 0*282 Ry for lithium), the estimated possible errors of ± 0*02 Ry at and ± 0*04 Ry at N become very appreciable. An interpolation scheme such as that used above would not have much significance in these circumstances.
560 J. F. Cornwell obtained agreed well with those of Herring & Hill (given in column 1 of table 5), except for the state K5. Calculations based on these levels, and carried out as outlined above, resulted in a Fermi energy of 0*90 Ry. The error in fitting the Ks level H S' K T Figure 3. Brillouin zone for the close-packed hexagonal lattice.